It becomes a lot more obvious when you consider two factors; 1, that every agent wants to maximize its utility, and 2, that no split will be agreed to if it makes one agent worse off for participating in the trade than their bull action. Consider our simplest factory case: a factory that requires 1 owner to supply capital and one worker to provide labor. Since both are required to obtain the output, you split the gains from trade by taking the income from the factory, paying out the costs to bring both agents to net 0 from agreeing to the trade (ie, maintenance on the capital, taxes, insurance against work injuries, food to replace calories expended in labor, etc). Then whatever is left, the profit, gets split between both.
Now expand this. If there is one supplier of Capital and 2 workers, but it still only needs 1 worker to function, how does the split go? If the workers have good decision theory, they realize their best option is to coordinate. The Capital provider and the Laborer each get 50-50 split, and since each laborer has a 50% chance to be chosen that comes out to 50-25-25. This comes from a few factors. First, the Capital owner won’t accept less than half. If he is better off refusing to deal with B entirely and just trade with A, obviously he will do that, so our split has to be at least 50 to him. However, the workers also won’t accept less than 25 in expectation, or else they are both better off with one of them going to the other and saying “I won’t compete with you in exchange for half your profit.” A 50% chance at a 70-30 split (since there is a half chance each is chosen) is worse than that in expectation, even if the nominal payout is higher. Thus if the owner won’t accept less than half and our workers won’t accept less than a quarter, the only possible split is 50-25-25. Since our position is symmetric, 2 Capital providers and one laborer get a 25-25-50 split. Assuming there are 2 of each, that comes down to 2 50-50 splits again, since any lesser split has a better alternative that one party can force. Thus in any trade that requires one member of party A to contribute and one of Party B, each member of party A gets 1/2*#A and each member of B gets 1/2*#B. If the factory requires 2 workers to function, you can treat those two workers as a single agent that distributes half of the profit between them.
This has a few desirable outcomes.
For one, you are never better off for having more competition
You are never worse off for having more people to trade with
There is no coordination between agents that can reduce this to a simpler form that gets in expectation a higher payout for that group
it is agnostic of which of a set of interchangeable options is chosen.
There is no incentive to create dummy agents
And most importantly,
it is very simple to calculate. However many workers there are split between them half of the profit, irrespective of how large the labor pool is. This gets them in expectation less money since they have less chance of getting picked, but doesn’t change the amount changing hands in each actual timeline.
First, I note that this comment thread was previously about Shapley values, and you don’t seem to have done any Shapley calculations. If this is meant to be the same rule, but explained from a different angle, then I don’t see how to establish equivalence. If this is meant to be a new system, then I don’t see how it generalizes to more complex examples, such as where the factory output scales with the number of workers (rather than being all-or-nothing). (I also don’t see why you’d choose this particular comment to start promoting your alternative system.)
Second, you’re analyzing a situation where a required input can be provided by any of multiple parties; that is, if there are 2 owners, you only need 1 owner to agree in order to make the factory run. But the story problem above was about a situation where you need all of multiple parties; i.e. replace the 1 owner with 1 capitalist + 1 technologist and you need both of them to make the factory run.
If I came to you and said, here’s a game with 3 people (1 worker + 1 capitalist + 1 technologist), you need all 3 people working together to produce anything, how do they split the profits? I suspect you’d say an even 3-way split. But that implies that the owner from the 2-person can divide himself into 2 dummy agents (1 capitalist + 1 technologist) and then demand 2⁄3 of the profit (up from 1⁄2) because he’s now (nominally) doing 2 out of 3 jobs.
How do you prevent this exploit?
Third, I don’t buy your claim in your advanced examples that “any lesser split has a better alternative that one party can force”. For instance, in the 1 owner + 2 workers example, if the owner offers worker A a 70-30 split, that’s better for both the owner and worker A than your proposed split, and I don’t see what worker B can do about it.
You seem to be arguing that worker A should reject this split on some sort of timeless logic (?) where A reasons that there was an equal chance the offer would have been made to B and so if A+B are the sort of people who accept this offer then they each get 15 in expectation across all counterfactuals. Even if you buy the timeless logic, this only works if A and B use correlated strategies such that A is effectively choosing for both of them; otherwise, after A rejects this split, the owner proposes it to B and A gets nothing in all counterfactuals. So that seems to me like a coordinated solution, not a solution that a single party can unilaterally force.
In fact, it looks to me like you’ve said something pretty close to “my system rewards monopolies, so A and B are incentivized to form a cartel and act like a single agent, and therefore I assume they do so.”
It becomes a lot more obvious when you consider two factors; 1, that every agent wants to maximize its utility, and 2, that no split will be agreed to if it makes one agent worse off for participating in the trade than their bull action. Consider our simplest factory case: a factory that requires 1 owner to supply capital and one worker to provide labor. Since both are required to obtain the output, you split the gains from trade by taking the income from the factory, paying out the costs to bring both agents to net 0 from agreeing to the trade (ie, maintenance on the capital, taxes, insurance against work injuries, food to replace calories expended in labor, etc). Then whatever is left, the profit, gets split between both.
Now expand this. If there is one supplier of Capital and 2 workers, but it still only needs 1 worker to function, how does the split go? If the workers have good decision theory, they realize their best option is to coordinate. The Capital provider and the Laborer each get 50-50 split, and since each laborer has a 50% chance to be chosen that comes out to 50-25-25. This comes from a few factors. First, the Capital owner won’t accept less than half. If he is better off refusing to deal with B entirely and just trade with A, obviously he will do that, so our split has to be at least 50 to him. However, the workers also won’t accept less than 25 in expectation, or else they are both better off with one of them going to the other and saying “I won’t compete with you in exchange for half your profit.” A 50% chance at a 70-30 split (since there is a half chance each is chosen) is worse than that in expectation, even if the nominal payout is higher. Thus if the owner won’t accept less than half and our workers won’t accept less than a quarter, the only possible split is 50-25-25. Since our position is symmetric, 2 Capital providers and one laborer get a 25-25-50 split. Assuming there are 2 of each, that comes down to 2 50-50 splits again, since any lesser split has a better alternative that one party can force. Thus in any trade that requires one member of party A to contribute and one of Party B, each member of party A gets 1/2*#A and each member of B gets 1/2*#B. If the factory requires 2 workers to function, you can treat those two workers as a single agent that distributes half of the profit between them.
This has a few desirable outcomes.
For one, you are never better off for having more competition
You are never worse off for having more people to trade with
There is no coordination between agents that can reduce this to a simpler form that gets in expectation a higher payout for that group
it is agnostic of which of a set of interchangeable options is chosen.
There is no incentive to create dummy agents
And most importantly,
it is very simple to calculate. However many workers there are split between them half of the profit, irrespective of how large the labor pool is. This gets them in expectation less money since they have less chance of getting picked, but doesn’t change the amount changing hands in each actual timeline.
First, I note that this comment thread was previously about Shapley values, and you don’t seem to have done any Shapley calculations. If this is meant to be the same rule, but explained from a different angle, then I don’t see how to establish equivalence. If this is meant to be a new system, then I don’t see how it generalizes to more complex examples, such as where the factory output scales with the number of workers (rather than being all-or-nothing). (I also don’t see why you’d choose this particular comment to start promoting your alternative system.)
Second, you’re analyzing a situation where a required input can be provided by any of multiple parties; that is, if there are 2 owners, you only need 1 owner to agree in order to make the factory run. But the story problem above was about a situation where you need all of multiple parties; i.e. replace the 1 owner with 1 capitalist + 1 technologist and you need both of them to make the factory run.
If I came to you and said, here’s a game with 3 people (1 worker + 1 capitalist + 1 technologist), you need all 3 people working together to produce anything, how do they split the profits? I suspect you’d say an even 3-way split. But that implies that the owner from the 2-person can divide himself into 2 dummy agents (1 capitalist + 1 technologist) and then demand 2⁄3 of the profit (up from 1⁄2) because he’s now (nominally) doing 2 out of 3 jobs.
How do you prevent this exploit?
Third, I don’t buy your claim in your advanced examples that “any lesser split has a better alternative that one party can force”. For instance, in the 1 owner + 2 workers example, if the owner offers worker A a 70-30 split, that’s better for both the owner and worker A than your proposed split, and I don’t see what worker B can do about it.
You seem to be arguing that worker A should reject this split on some sort of timeless logic (?) where A reasons that there was an equal chance the offer would have been made to B and so if A+B are the sort of people who accept this offer then they each get 15 in expectation across all counterfactuals. Even if you buy the timeless logic, this only works if A and B use correlated strategies such that A is effectively choosing for both of them; otherwise, after A rejects this split, the owner proposes it to B and A gets nothing in all counterfactuals. So that seems to me like a coordinated solution, not a solution that a single party can unilaterally force.
In fact, it looks to me like you’ve said something pretty close to “my system rewards monopolies, so A and B are incentivized to form a cartel and act like a single agent, and therefore I assume they do so.”